It was numerically demonstrated in Refs. [1]}, [2]}, [3]} that for the case where the time between measurements \(\tau \)  is small (compared to typical time scales in the unitary evolution) the above dynamics is accurately described by a continuous time evolution with a non-Hermitian effective Hamiltonian. However, to obtain a description with continuous time, we need the limit \(\tau \rightarrow 0\)  and this is problematic because it leads to the Zeno effect.
Here we discuss a systematic approach where this problem is overcome and the effective non-Hermitian description
can be obtained in a rigorous way. For this we let the system-detector couplings scale with the detection interval \(\tau \) 
as
\(H_{i\alpha }^{({\mathcal {SD}})}=\sqrt{\frac{\gamma _{i\alpha }}{\tau }},\,\,\,H_{\alpha i}^{({\mathcal {SD}})}=\sqrt{\frac{\gamma _{\alpha i}}{\tau }}.\) 
